Limiting partition function for the Mallows model: a conjecture and partial evidence

TL;DR

This paper investigates the asymptotic behavior of the partition function in a class of Gibbs models on permutations, proposing a conjecture linking it to Fredholm determinants and providing partial evidence.

Contribution

It introduces a conjecture relating the limit of the partition function to Fredholm determinants and offers partial evidence supporting this connection.

Findings

Conjecture that the limit is given by a Fredholm determinant.

Partial evidence supporting the conjecture.

Identification of a connection to Schr"odinger bridge distributions.

Abstract

Let $S_n$ denote the set of permutations of $n$ labels. We consider a class of Gibbs probability models on $S_n$ that is a subfamily of the so-called Mallows model of random permutations. The Gibbs energy is given by a class of right invariant divergences on $S_n$ that includes common choices such as the Spearman foot rule and the Spearman rank correlation. Mukherjee in 2016 computed the limit of the (scaled) log partition function (i.e. normalizing factor) of such models as $n\rightarrow \infty$. Our objective is to compute the exact limit, as $n\rightarrow \infty$, without the log. We conjecture that this limit is given by the Fredholm determinant of an integral operator related to the so-called Schr\"odinger bridge probability distributions from optimal transport theory. We provide partial evidence for this conjecture, although the argument lacks a final error bound that is needed for it to become a complete proof.